Functions; If you were to ask five different people what they thought a function was, you would probably get six different answers, and that’s O.K. Functions are one of those things I had to wrap my head around quite a few times till I derived a simple way of thinking about it. First, I will list a couple of ways you can define and think about a function, then I will attempt to simplify the concept so that there is no need to memorize a definition, you will simply just understand and that’s the best way, really.
Function Definitions
- A function connects one set (input) to another set (output)
- A function takes an input and gives only one output
- A functions output compared to its input is an ordered pair
- A function is a notation (concept) for solving a problem
- A function is derived from the idea that if two things are equal in math, you can substitute for the other f(x)=y
- A function is a “relationship” that maps 1 input to 1 output
- A function; f(x)=y defines 1 variable in terms of another. A function is a rule or a step by step process such as an algorithm relating y to f(x)
Proportional Functions
There are many types of functions, but the most basic types are; proportional functions which are linear functions. Proportional functions are defined by the equation y=kx
Proportional functions are functions where the output is equal to the input multiplied by the constant. y=kx
In this proportional equation, K = the constant, X = the input, and Y = the output.
Other operands names commonly used for X is; Independent variable and Y is called, a Dependent variable. (there are others operands names used like Domain and Range but that’s for computer science)
Let me try now to simplify; I like to think of a function analogous to a married couple; a Husband and a Wife and a Child. The Husband would be the constant (k) because he is the rock of the family, the Wife is the Independent variable (x) because she is allowed to change, and the Child is the Dependent variable (y) because it “Depends” on the parents. So kx=y. In this equation the k is the “constant of proportionality” because it’s the value that relates the two amounts.
Example 1: you are paid $40 an hour
How much you earn is directly proportional to how many hours you work
Work more hours, get more pay; in direct proportion.
This could be written:
Earnings ∝ Hours worked
- If you work 2 hours you get paid $80
- If you work 3 hours you get paid $120
- etc …
The constant of proportionality is 40 because:
Earnings = 40 × Hours worked
Example 2: Remember another way of thinking of a function is as a “rule” it maps 1 input to 1 output. Consider the following question;
How many tires are there in a parking lot of 5 cars, where each car has 4 tires?
y = F(x) = 4(x) = y
Using an xy table you can see that there are 20 tires.
Example 3: Earlier I mentioned that a proportional function was a linear function, this is true but it is important to ask the question;
Are all linear functions proportional functions?
“Linear functions have a constant rate of change and describe a straight line on a graph. Linear functions can be defined by the equation y = mx + b. Proportional functions are linear functions that include the origin and can be defined by the equation y = kx. All proportional functions are linear, but not all linear functions are proportional.”
Consider this linear function has the following form
y = f(x) = a + bx
A linear function has one independent variable and one dependent variable. The independent variable is x and the dependent variable is y.
a is the constant term or the y intercept. It is the value of the dependent variable when x = 0.
b is the coefficient of the independent variable. It is also known as the slope and gives the rate of change of the dependent variable.
Proportional Function Wheels
I have been a big fan of proportional function wheels (or at least that is what I call them, I like to think I coined the phrase) ever since I was introduced to them back in school as a boy. Of course you can always transpose the equation but I like the mental visualization associated with function wheels.
Distance, Velocity , Time
Mass, Density, Velocity
Circumference, π, Diameter
Total Rise, Riser, Step
Another benefit to using proportional function wheels; they make it pretty easy to remember equations and their proportionality. For example; the above DVT wheel, D is Directly proportional to V and likewise D is directly proportional to T but V is inversely proportional to T. Can you see the pattern? Basically the top variable and the bottom variable will always be Directly proportional and the two bottom variables will be inversely proportional.
Understanding this relationship makes it easier to understand seemingly wrong solutions because they are contradictory. For example, there is a contradiction between ohm’s law and watts law where in ohm’s law current (I) is directly proportional to voltage (E) but in watts law they are inversely proportional. But why is this…? Are any of the laws wrong? Of course not! They wouldn’t be laws if they were. They both are correct. Ohm’s law is correct that current (I) is directly proportional to voltage (E) in the context of Resistance (R). I ∝ E at a constant resistance. Watts law is also correct in that current (I) is inversely proportional to Voltage (E) in the context of Power (P). I ∝ 1/E at a constant Power.
